hyperspectral image compression using a two-stage klt

HYPERSPECTRAL IMAGE COMPRESSION

USING A TWO-STAGE KLT

A Thesis

presented to

the Faculty of California Polytechnic State University,

San Luis Obispo

In Partial Fulfillment

Of the Requirements for the Degree

Master of Science in Electrical Engineering

by

Seton S. Schroeder

May 2009

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© 2009 Seton S. Schroeder

ALL RIGHTS RESERVED

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COMMITTEE MEMBERSHIP

TITLE: Hyperspectral Image Compression using a Two-stage KLT

AUTHOR: Seton S. Schroeder

DATE SUBMITTED: May 1, 2009

COMMITTEE CHAIR: John A. Saghri, Associate Professor

COMMITTEE MEMBER: Fred W. DePiero, Professor

COMMITTEE MEMBER: Jane Zhang, Assistant Professor

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ABSTRACT

Hyperspectral Image Compression using a Two-stage KLT

Seton S. Schroeder

A computationally efficient, lossy, bandwidth compression scheme for hyperspectral imagery is presented. In certain cases, the direct application of the standard KLT algorithm over an entire data set is not practical. The data may be too large for efficient processing by one encoder and so portions of the entire data set may necessarily be sampled by different encoders. The algorithm proposed handles such a limitation by leveraging the standard JPEG 2000 technology. The component decorrelation of the JPEG 2000 (extension 2) is replaced with a two-stage Karhunen-Loeve Transformation (KLT) operation resulting in a reduction in the computational complexity. At the first stage the spectral data is partitioned into sets of adjacent spectral images, which are decorrelated independently of one another. At the second-stage inter-partition correlation is exploited by combining principle component images and performing a second set of KLT operations. It is shown that reconstructed image quality is improved by minimizing partition size at the first stage. The method is evaluated using the criteria of root-mean-square error as well as the performance of a terrain-classification algorithm. A relationship between partition size and minimal inter-partition decorrelation is also proposed and may act as a guide in the tuning of compression settings.

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TABLE OF CONTENTS

List of Tables ………………………………………………………………………...… vi List of Figures ………………………………………………………………..……..… vii Chapter 1 Introduction ………………...……………………………………………… 1 Chapter 2 Background …………………...……………………………………………. 3

2.1 Multispectral Images ……..……….……………………………..……………… 3 2.2 KLT in Compression and Transmission of Multispectral Images………………. 4

Chapter 3 Recent Data-compression Approaches …………..……………………….. 7 3.1 Bidirectional Interband Prediction Method ……………..………………………. 7 3.2 Distributed Karhunen-Loeve Transform …………………..……………….…… 9

3.2.1 Local KLT Constraint ……………...……………………………………...…... 11 3.2.2 Distributed KLT Algorithm ……………………………………………………. 15

3.3 Sub-optimal KLT ……………….………………………………………........... 15 Chapter 4 Thesis Methods ……………………………...………………………….… 18

4.1 Spectral Compression Technique ……………...……………………………..... 18 4.1.1 Protocol ………………………………………..………………………………… 19 4.1.2 KLT Algorithm ………………………………………………………………….. 20

4.2 Spacial Compression Technique ……………………………………………….. 22 4.3 Reconstruction …………………………………………………………………. 24 4.4 Classification …………………………………………...…………………....… 24 4.5 Quantitative Assessment of Reconstruction ………….………..………………. 25

4.5.1 Root-mean-squared Error ………………………………...…………………… 26 4.5.2 Confusion Matrix ………...………………………………...…………………… 26

4.6 PC Contribution to Second-stage KLT ……………….………..………………. 27 Chapter 5 Results ……………………………………………………………………... 29 Chapter 6 Conclusion and Future Work ………………………………………….… 37 Bibliography ……………………………………..……………………………………. 47 Appendix …………………………………………….……………………………….... 48

The Karhunen-Loeve Transform ……….. ……………….………..………………. 48 Concepts …………………………………………………………...…………………… 48 Derivation …….…...………………….………...………………...…………………… 51

Matlab Code ……….………….. ………. ……………….………..………………. 54

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LIST OF TABLES Table 5-1 8 File/group Confusion Matrix ……………………………………………… 34 Table 5-2 16 File/group Confusion Matrix ……………..……………………………… 34 Table 5-3 20 File/group Confusion Matrix ……………..……………………………… 35

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LIST OF FIGURES

Figure 2-1 The Electromagnetic Spectrum ……………………………………….……... 4 Figure 2-2 Spectral Image Compression and Reconstruction ……………………..…….. 6 Figure 3-1 Marginal KLT ……………………………………………….......…..…..…. 10 Figure 3-2 Local KLT ………………………………………………………………….. 13 Figure 3-3 Sub-optimal KLT …………………………………………………………... 17 Figure 4-1 Two-stage KLT Operation …………………………………………………. 20 Figure 5-1 Spectral-grouping Effect on Image Degradation ….………………….……. 29 Figure 5-2 Ground Truth Thematic Map …………………………………………….… 30 Figure 5-3 8 File/group Thematic Map ……………………………………...……….… 31 Figure 5-4 16 File/group Thematic Map ………………………………………….….… 32 Figure 5-5 20 File/group Thematic Map ……………………………………………..… 33 Figure 5-6 Second-stage Decorrelation Analysis ….………………………………...… 36 Figure 6-1 Partitioning and Encoding in JP3D ……………………………………….... 44 Figure 6-2 JP3D scanning Pattern ……………………………………………………… 44 Figure A-1 KLT Conceptual Example …………………………………………….…… 49 Figure A-2 Effect of Spectral Axis Rotation on Data Variance…………………...…… 52

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Chapter 1 Introduction

Imaging technology has made it possible to extract a great deal of information

about the content of the Earth’s surface and atmosphere. One of the most advanced

systems, the hyperspectral imager, has allowed the capture of electromagnetic waves over

the visible and infrared regions, with high spectral resolution. The motivation for the

study of hyperspectral image processing lies in the number of useful applications the field

has. In agriculture, it aids in the identification of soil composition. It is used in

environmental studies to determine the degree of land and air pollution, as well as forest

monitoring. Hyperspectral imaging also has uses for the military in surveillance. Many of

these applications require gathering information of expansive regions. Therefore, aircraft

and satellites are often the platforms used to capture such large scene. However the

bandwidth required to transmit the information to a ground station on Earth is currently

not sufficient for the image data in its raw form. Therefore, compression techniques that

attempt to further optimize the quality and usability of transmitted data continue to be

investigated and used.

The purpose of this project was, firstly, to review a few of the most current

multispectral image compression techniques. The second aim was to propose a

computationally simpler alternative that can handle the restrictive scenario where the

spectral images have been partitioned and compressed at different terminals. A

quantitative goal was to compress 144, 256x256 pixel spectral images at 0.25 bpp such

that the reconstructed images exhibited a root-mean-squared error of less than 1%.

Looking at variations in image reconstruction quality due to changes in spacial

compression ratios, however, was not an aim of this project. A second goal of this thesis

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was to receive a total classification accuracy of greater than 95% when applying the

reconstructed image to a 6-class, minimum Euclidean distance, general hill climb,

unsupervised classification scheme. Iterations were stopped after a 1.8% change in pixel

classification was achieved. This produced less than10 iterations in every case.

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Chapter 2 Background

The data processing performed in this project made use of images of reflected

electromagnetic wave information, covering a range of frequencies. The first step

involves a typical procedure of spectrally decorrelating data and removing spectral

redundancies. Once the data is spectrally compressed, the images are converted to a

format that removes spacial redundancies. Finally, the images are decompressed and the

reconstructed image data are compared, pixel-by-pixel, to the original spectral image

data, using a root mean squared error calculation.

2.1 Multispectral Images

One technique for gathering information about the content of atmosphere or land

is through the data provided by multispectral images. Imaging systems have made it

possible to capture the radiant energy that is reflected and emitted from the air and land

cover. This energy is carried on the various wavelengths of the electromagnetic spectrum.

The spectrum can be divided into 7 bands. Ranging from lowest wavelength to highest

wavelength, these bands are: Radio waves, Microwaves, Infrared, Visible, Ultra Violet,

X-rays, and Gamma rays (Figure 2-1). Human eyes can only detect energy delivered

from an object at wavelengths that fall within the Visible Light spectrum, which ranges

from 0.4 µm (the color violet) to 0.7 µm (the color red). Special sensors, however, allow

information to be captured at wavelengths outside this range. Hyperspectral imagers

specifically detect the Visible, Near Infrared, Mid Infrared, and Thermal regions. The

images captured by such technology often then go through a compression scheme such as

the Karhunen-Loeve Transform, before they can be transmitted.

Figure 2-1 The Electromagnetic Spectrum. Spectral images contain information carried by electromagnetic waves categorized by their 2.2 KLT in Compression and

The Karhunen-Loeve Transformation (KLT) operation is the foundation for the

spectral compression method used in this project. It is a standard technique used to

compress and separate a scene’s multiple spectral images

images of uncorrelated data

Appendix for a conceptual description and mathematical derivation of the process.

The process of converting spectral images of a scene into its principle components

using the KLT only compresses information that varies with spectral frequency.

However, before such information

spatially compressed as well. Joint Photographic Experts Group (JPEG) is an

organization that developed a 2

[2].

When images are sent in a compressed JPEG 2000 format from a satellite to Earth

they are then decompressed. The process of decompressing the image data involves first

converting the data back to its raw data format via the inverse JPEG 2000 operation. This

puts the data into a form that can allow for spectral reconstruction of the original images.

An inverse KLT operation is performed to bring the principle component images back to

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The Electromagnetic Spectrum. Spectral images contain information carried by electromagnetic waves categorized by their wavelengths [1].

KLT in Compression and Transmission of Multispectral Images

Loeve Transformation (KLT) operation is the foundation for the


separate a scene’s multiple spectral images into ‘principal component’

images of uncorrelated data. To do such, linear algebra techniques are utilized.



the KLT only compresses information that varies with spectral frequency.

However, before such information can be transmitted over limited bandwidths, it must be

lly compressed as well. Joint Photographic Experts Group (JPEG) is an

organization that developed a 2-D image compression standard to do just that, JPEG 2000






The Electromagnetic Spectrum. Spectral images contain information lengths [1].

Transmission of Multispectral Images

Loeve Transformation (KLT) operation is the foundation for the


into ‘principal component’

. To do such, linear algebra techniques are utilized. See the



the KLT only compresses information that varies with spectral frequency.

bandwidths, it must be

lly compressed as well. Joint Photographic Experts Group (JPEG) is an

D image compression standard to do just that, JPEG 2000






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their spectral-image form. If all the image data is used in the compression, transmission,

and reconstruction process then, in theory, no information should be lost. In practice, this

would require no round-off and/or quantization errors, in the computational process. Such

perfect reconstruction is known as lossless transmission. If only a subset of the principle

component images are transmitted, the process is lossy, and the reconstructed image will

not contain all the original image information. Both KLT and JPEG 2000 were used in

this project to spectrally and spacially compress and decompress the image data in a lossy

fashion (Figure 2-2). But before further describing the methods used in this project, a

review of some other recent data-compression approaches will be analyzed to set a basis

for the evaluation of this thesis’ approach.

Figure 2-2 Spectral Image Compression and Reconstruction. The spectral images are compressed and reconstructed using KLT and JPEG operations.

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Spectral Image Compression and Reconstruction. The spectral images are compressed and reconstructed using KLT and JPEG operations.

Spectral Image Compression and Reconstruction. The spectral images are compressed and reconstructed using KLT and JPEG operations.

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Chapter 3 Recent Data-compression Approaches 3.1 Bidirectional Interband Prediction Method

Ideally, it is best to use all the pixel data of one’s spectral images for a particular

scene, however transmitting all the data is not feasible due to downlink channel capacity.

Therefore, one approach is to assume a linear relationship between the intensities (due to

reflection and/or emitted energy) of particular bands, in order to predict other pixel

values. If one were to have a limited set of spectral images, one could estimate the

parameters that describe the linearity between each two-band associations in a set. For

example, if one were to observe the intensities of the same pixel from two spectral

images of a ground scene, the relationship between these intensities could be written as

such:

I1(p1) = aI2(p1) + b (1) ,

where I1 and I2 are the intensities of a pixel, p1, for two arbitrary bands [3]. The

parameter, a, describes the rate at which I1 changes as I2 changes for the given feature.

The parameter, b, can typically be described as the average difference between the

intensity of the two bands for the given feature. To make computation easier, one could

determine a set of parameters for equally-sized block partitions of a whole scene. The

larger the size of the blocks, the lower the overhead associated with the transmitted

prediction parameters. The tradeoff, however, is a larger prediction error - especially with

busier scenes.

This linear model is straight forward, however what is not, is the determination a

reference band, from which predictions of one of the other bands will be made. In other

words, one would have to decide on which spectral image’s intensities to take, I2, from

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which to make a prediction of the spectral illumination, I1, of a pixel at another

wavelength . An approach to this is to simply take the band that yields the minimum sum

of prediction error variances when a best-fit linear relationship is made with all the other

bands, of all blocks, for all pixels. Simulations have demonstrated that using a middle

wavelength as a reference in either the visible region or infrared region provides a better

prediction for other wavelengths [3].

However, there are downsides to this linear prediction method. The coding noise

accumulates significantly at lower bit rates, causing coding distortion in the anchor band,

reducing the correlation accuracy with the prediction band. Also, using only one band as

a reference is not always an effective enough constraint for predicting the intensity levels

of another band. Therefore, a method that uses two reference bands to predict a single

prediction band was developed, which solves both of the issues just described [3].

The modified linear equation that makes use of two reference bands is similar to

the single-reference band equation, but with an additional scaling term:

I1(p1) = aI2(p1) + bI3(p1) + c (2) ,

where the prediction band intensity is a sum of weighted contributions from two

reference bands, in addition to a constant offset, c. In this new scenario each prediction

band for each block is associated with two particular reference bands. Depending on the

location of the prediction band, one of the anchors is a band that falls in the middle of the

Visible or Infrared regions. The second anchor band falls on the other side of the

prediction band, so that the prediction band is flanked. This second band is determined by

starting at the lowest available band, if the prediction band is higher, and moving up one

band at a time band. If the prediction band is lower than the second band, then the

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iterations of second band choices starts at the highest possible band. For each secondary

reference band chosen, a root mean squared error is determined. When the error reaches a

local minima, iterations cease and the current reference bands at that iteration are used.

The bidirectional prediction scheme is very effective at providing better-quality

reconstructed images while adding little additional parameter data as overhead. The use

of two reference bands minimizes further the mean square error values between the

original and reconstructed images without much more overhead. However, this method

requires that, for every band prediction made, a whole set of linear equations be

determined. Furthermore, each band is processed separately. This can be very impractical

for a large number of spectral bands. The Visible band and Infrared band references that

optimize the reconstruction can vary as well, depending on the number of bands used.

Finally, the process of finding the second reference band, at best, provides a local minima

error. However, a method based on heuristics can often do better (e.g. using the median

visible and infrared bands as anchors and using sample pixels in the calculation of a

maximum covariance matrix for the selection of the second reference band).

3.2 Distributed Karhunen-Loeve Transform

Applying the Karhunen-Loeve Transform to distinguish correlating bands,

although just as good as a bidirectional interband prediction method, can carry much

more computation and overhead [3]. This is particularly true when one attempts to apply

a single KLT operation to the entire set of a very-large number of bands, describing a

whole scene. When applied in this manner, the KLT is known specifically as a direct,

full, or joint KLT. This method may also not even be possible if different band regions

are fed into separate, non-communicating terminals, before being collected into a fusion

center. The terminals perform compression and encoding only on the sub

images that they receive.

processed together. A simple alternative, in this scenario, would be to apply a separate

KLT operation to each set of spectral images for each terminal. This local approach to the

KLT strategy is known as a

however, is that it completely

terminals. The

Figure 3-1 Marginal KLT. One approach to a KLT operation onspectral images is to perfotransformed images are passed thrat their own bit rates, T

operation only makes use of local statistics. Given the constraints of a partitioned

processing method for a set of spectral images, an optimal KLT approach would lie

somewhere between the ideal

Such an approach, which provides the information of reconstructed spectral images to

those spectral images that have yet to be processed, is termed the

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The terminals perform compression and encoding only on the sub

images that they receive. They are transmitted to their respective receivers an

A simple alternative, in this scenario, would be to apply a separate


KLT strategy is known as a marginal KLT (Figure 3-1). The downside to

however, is that it completely ignores the correlation between bands detected by separate

KLT. One approach to a KLT operation on large number of spectral images is to perform it on groups of spectral images, xi, separately. The transformed images are passed through individual encoders and spacially compress at their own bit rates, Ti [5].



somewhere between the ideal joint KLT and the oversimplified marginal

an approach, which provides the information of reconstructed spectral images to

those spectral images that have yet to be processed, is termed the distributed

The terminals perform compression and encoding only on the sub-spectrum of

They are transmitted to their respective receivers and then

A simple alternative, in this scenario, would be to apply a separate


1). The downside to this method,

detected by separate

large number of , separately. The

ough individual encoders and spacially compress



marginal KLT technique.

an approach, which provides the information of reconstructed spectral images to

distributed KLT [3].

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A distributed KLT algorithm that minimizes the approximation error or distortion

rate of compressed and reconstructed images can be derived if one looks at a particular

constraining scenario. The scenario that will be investigated is one where all but one of

the terminals has a fixed transform matrix and the goal is to optimize the transformation

matrix of the remaining terminal given the eigenvectors of the others, yielding the

optimal local KLT [3].

3.2.1 Local KLT Constraint

As with a marginal KLT approach, the local KLT scenario looks at an individual

terminal and performs a KLT. The difference is the conditionality of the parameters used

in the KLT. The transformation matrix of the marginal KLT and eigenvalues are affected

only by local statistics. In a local KLT operation, the eigenvalue and eigenvector values

of the isolated terminal are determined from the results of the transformation parameters

of the other terminals. The parameters of the transformation at the isolated terminal are

chosen such that the total mean squared error after reconstruction, is minimized. The

method for minimizing this error is called linear approximation.

A simple solution for determining the best linear approximations for every

terminal, in the marginal KLT case, does not seem to exist in general [5]. Therefore, the

local KLT scenario is a next-best approach. When it comes to the spacial compression of

multiple terminals in the marginal KLT scenario, the rates of compression, which are

related to the local eigenvalues, are independent of each other. This independency does

not make for fully optimized compression rates either. Unfortunately, the optimal

compression rates for a multiple-terminal scenario is mostly unknown to date, except in

the case of the two-terminal case [5]. Therefore, in the local KLT scenario one can divide

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the spectral images into two partitions. One partition simply contains the spectral images

of the terminal whose linear approximation we are trying to optimize. The second

partition consists of the spectral images from all the other terminals together. The

terminals of this second partition are given user-defined compression rates from which

the isolated terminal’s compression rate will be determined. However, this all depends on

the first step of creating two spectral partitions, which will now be described in more

detail.

Once the spectral images are partitioned, a new covariance matrix can be created,

which segments the covariance matrix of the full set of spectral images. The new

covariance matrix can be seen best in its matrix form:

Cx’ = � �� (3) ,

where Cx1 is simply the local covariance matrix of the single-terminal partition.

Likewise, Cx12 is the set of covariances between every spectral value from the single-

terminal partition and every spectral value from the multi-terminal partition. The

covariance matrices are defined in the same manner. Furthermore, the multiple-terminal

partition’s linear approximation can be expressed as:

y2 = Ρ2x2 + z2 (4) ,

where x and y are vectors that denote a pixel’s original spectral values and principle

component values, respectively [5]. Ρ2 is the matrix of eigenvectors, arranged in rows, of

the covariance matrix for the multi-terminal partition. The vector, z2, is the Gaussian

random noise associated with the chosen linear approximation component images’ pixel.

The goal is to optimize the transformation matrix of the isolated terminal such that the

distortion of the reconstructed spectral images, ��, is minimized (Figure 3-2).

Figure 3-2 Local KLT. Another approach to performing a KLT number of spectral images is to partition the spectral images into two groups, xx2. The transformation parameters for one the other [5]. In order to derive

covariance matrix is constructed. This matrix i

partitioned covariance matrix,

A = [ Cx21 Cx2Ρ2’ ] A is a matrix with number of rows equal to the number of spectral bands originally

grouped into the multi-terminal partition. Its number of columns equals the number of

original spectral bands in the single

the multi-terminal partition. Let

of original spectral bands in the single

covariance matrix for our linear approximati

Cw’ = [ Cx1 – C

This equation can be written in its simpler form as: Cw’ = Qw’Dw’Qw’’

Ρ1

Ρ2

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2 Local KLT. Another approach to performing a KLT operation to a large number of spectral images is to partition the spectral images into two groups, x

parameters for one partition is fixed and used

In order to derive a new set of optimal eigenvectors for the local KLT, a new

covariance matrix is constructed. This matrix is associated with the values of the

partitioned covariance matrix, Cx’. We start by considering a matrix A, defined as:

] -1 (5) .

is a matrix with number of rows equal to the number of spectral bands originally

terminal partition. Its number of columns equals the number of

original spectral bands in the single-terminal partition plus the number of

terminal partition. Let A1 consist of the first set of columns equal to the number

of original spectral bands in the single-terminal partition, M. We can then define the new

covariance matrix for our linear approximation as:

Cx12Ρ2’(Ρ2Cx2Ρ2’+Ρz)-1 Ρ2Cx12’ ][ IM A1’ ]

This equation can be written in its simpler form as:

(7) .

operation to a large number of spectral images is to partition the spectral images into two groups, x1 and

partition is fixed and used to optimize

KLT, a new

values of the

, defined as:

is a matrix with number of rows equal to the number of spectral bands originally

terminal partition. Its number of columns equals the number of

e number of eigenvectors in

consist of the first set of columns equal to the number

. We can then define the new

(6) .

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One will notice the similarity of this equation to that defining the relationship between

the covariance matrix of spectral values of an image and its associate eigenvectors,

described under the KLT derivation found in the Appendix. Recall,

CX = EDE’ (8) ,

where CX was the covariance matrix, E was the eigenvector matrix (written as columns),

and D was a diagonal matrix describing the eigenvalues. In fact, in our new local KLT

algorithm, the eigenvectors are the columns of Qw and the eigenvalues are the diagonal

values of Dw. We relate the original single-terminal partition’s transformation matrix to

Qw and A1 as follows:

Ρ1 = [ Qw(M) ]� &'(� � (9) ,

where Qw

(M) refers to the first M columns of Qw and IM is an M dimensional identity

matrix.

To determine, in practice, the most appropriate transformation parameters to use

in the local KLT scenario, one must have access to y2 ( or Ρ2x2 + z2 ). A constraint is

that the best linear approximation is dependent on the values used for y2. The scheme for

optimizing spacial compression, for the local KLT case, is similar to the method for

spectral compression. One chooses an arbitrary fixed rate for our multi-terminal partition,

and use this to determine an optimal rate for the single-terminal partition [5]. There does

not seem to be any simple, direct solution to this local KLT approach, however,

achieving local minima root-mean-squared errors have been demonstrated. Furthermore,

the method has been proven to be more effective than a marginal KLT approach [5].

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3.2.2 Distributed KLT Algorithm

The distributed KLT approach is an improvement on the local KLT constraint. It

is similar in the sense that the linear approximation of one partition is optimized, while

the KLT operation on the rest of the spectral data is held fixed [5]. However, the

distributed KLT separates the terminals into more than two partitions. One partition acts

as the single-terminal partition, as described in the local KLT scenario, and the

transformation matrices of all the other partitions are not changed [5]. The distributed

KLT method is performed under this situation, but using only a subset of pixels from the

scene image. After optimization of the single-terminal partition’s transformation matrix, a

single pixel is added to the sample and a new ‘single-terminal partition’ is chosen before

a local KLT operation is performed again. Every time a new pixel is added to complete

the scene, a different partition’s linear approximation is optimized. As the number of

pixels used in the local KLT operation increases, the linear approximations of each

partition will improve. The chances of reaching a local minima is dependent on the

random choice sequence of which terminal to optimize. However, the iterations can be

performed multiple times to the scene image until a local minima is found. Optimal

spatial compression rates for each partition can be computed in a similar fashion.

3.3 Sub-optimal KLT

Another simpler approach to addressing the correlations between groupings of

spectral bands is to apply the sub-optimal KLT [8]. In this method, the original spectral

images are divided into groups of the same number of images. The number of files is

arbitrary but allows for an easy computation of the KLT for each group. A subset of the

principle component images associated with the highest eigenvalues, are then combined

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into sets of new groups, and a second stage of KLT operations is performed. The unused

first-stage principle components are sent directly to the encoder for transmission.

Once a second-stage set or principle components are generated, the principle

component images of non-combined spectral images are combined. As before, images of

larger eigenvalues are paired and passed through yet another KLT operation. The same is

done with the lower eigenvalue images. The process continues up to an arbitrary point.

This process can be performed by breaking up groups into pairs of 8 images each (Figure

3-3). The division of the KLT groups into two-level sub-blocks of three KLT operations,

is performed because it has been shown that such an architecture yields insignificant

degradation in the coding gain for a large number of spectral images (0.3 % for N = 64).

However, this assumes that the signals used reflect a Markov process [8]. Still, such an

approach allows for more thorough removal of spectral redundancy across the entire

spectrum of images, and therefore greater compression with minimal effect on coding

gain.

Figure 3-3. Sub-optimalat multiple stages to make use of the correlation between spectral information in groups along the entire spectra of images

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optimal KLT. This method interleaves principle component images at multiple stages to make use of the correlation between spectral information in groups along the entire spectra of images [8].

interleaves principle component images at multiple stages to make use of the correlation between spectral information in

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Chapter 4 Thesis Methods

This thesis project involved a data compression scheme similar to the sub-optimal

KLT approach. However, it addresses the interband correlation issues highlighted in the

development of the distributed KLT algorithm. The bidirectional interband prediction

approached was not incorporated, but helped to support the notion that the grouping of

spectral image patterns shared by multiple spectral images can lead to better

reconstruction results upon decompression.

The spectral image processing, in the case of this project, involved images taken

of a region of San Francisco. Raw hyperspectral images of the scene, taken by a HIRIS

(Hyperspectral Infrared Imaging System) instrument onboard an airplane, were utilized

[9]. Each image provided the reflected illumination from the city for a very particular,

narrow, electromagnetic bandwidth. These multispectral images were spectrally

compressed using a very particular computationally-effective KLT algorithm written in

Fortran. This was followed by spacial compression using the JPEG 2000 standard. The

images were then decompression and reconstructed so that the resulting and original

spectral images could be quantitatively compared. Statistical calculations and a feature

classification process were performed using Matlab.

4.1 Spectral Compression Technique

The 144, 256 x 256 pixel, spectral images were AVRISS overhead the city [9].

The compression algorithm used in this project involved a two-stage KLT operation.

There are many computational algorithms that can achieve what the KLT operation tries

to accomplish. In this paper, one of the simpler, but effective, algorithms was applied.

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4.1.1 Protocol

The protocol for using the KLT operation, firstly, consisted of separating all the

files into groups of spectrally adjacent images. Each group contained the same number of

images. An exception to this is if the number of images chosen for each group was not

divisible by 144, in which case one group would contain fewer images. Secondly, the

KLT operation, described in Appendix, was performed separately on each group of the

images, one group at a time. This process of converting all the spectral images to their

principle component images was performed three times. Each time the number of images

used in a group was different. Results were observed for: 20, 16, and 8 files per group.

Once the KLT operation was applied, inter-group correlation had to be addressed.

An approach was therefore taken in which the groups of resulting principle component

images were interleaved and regrouped so that all the first principle component images

were put into one group, all the second principle component images were put into another

group, and so forth, down to four principle components deep. At this point, the KLT

operation was performed once again, however this time, on each group of principle

component images, one group at a time (Figure 4-1).

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Figure 4-1. Two-stage KLT Operation. The first-stage KLT operation is performed on the ‘m’ groups of ‘n’ spectral images each, X((i-1)n+1) – X(in), where i = 1,2,3… m. This results in ‘m’ sets of PC images. The PC images are then regrouped by level (1-4 in this example) and each of these groups, of ‘m’ PC images each, goes through a second KLT operation. The result is sets of second-stage PC images, Y, where the number of sets is equal to the depth of first-stage PC images that were regrouped for second stage transformation. 4.1.2 KLT Algorithm

The objective of the KLT is clear if one understands the operation conceptually

and linear algebra proofs being utilized. However, executing the operation,

computationally, is not so straightforward. Perfect results cannot be achieved, therefore a

number of algorithms have been designed to get ‘good enough’ approximations.

21

As the KLT derivation in the Appendix indicates, the covariance matrix of our

spectral values, X, can be expressed as:

CX = EDET (1) ,

where one tries to find the appropriate eigenvector values, E, that will diagonalize D.

While real non-symmetric matrices can be diagonalized, the transformation matrix is not

necessarily real [7]. However, a real similarity transformation can get close to doing the

job. It can reduce the matrix to many two-by-two blocks along the diagonal and leave the

other elements 0. The strategy is to iteratively flank the diagonal matrix, D, with a series

of transforms until the dot product of these transforms, on either side, bring D close

enough to a diagonal matrix. The inner product of the transforms will then yield the

eigenvectors. The transform right multiplying with D is the inverse of the transform left

multiplying with D. The resulting matrix is a rotated version of the original D. This result

is then flanked again by another transform and its inverse in the same manner and a new

rotated D matrix is computed. This process continues until the final result, CX, is a

diagonal matrix, within computer precision [7].

There are basically two general approaches to iteratively diagnolizing the matrix,

D [7]. Most modern eigensystem algorithms use both in combination for better results.

The first approach is to design transformation matrices to zero out a particular off-

diagonal element or row/column of D. However, a finite sequence of simple

transformation matrices is not generally enough to completely diagonalize a matrix to

within machine precision. Therefore, one can iterate the finite sequence of transforms

until the off-diagonal values are negligible. Otherwise one can make use of a second

approach, the factorization methods. In this approach the covariance matrix can be split

22

into the inner product of a left and right factor [7]. The left factor is the transformation

matrix and will satisfy the equation:

FRFL = FL-1DFL (2) ,

The matrix diagonalization approach used in this thesis is known as the Jacobi

Transformation algorithm. As described above, the simple sequence of transforms is each

a rotation designed to zero-out one of the off diagonals. Successive transformations undo

previously set zeros, but overall the off-diagonal elements get closer to zero. The Jacobi

method is a simple method compared to many others and is more appropriate for matrices

not much larger than 10. The computational processing time becomes significantly larger

after this.

4.2 Spacial Compression Technique

The result of all the KLT operations was a number of second-stage KLT principle

component images and a left-over set of first-stage KLT principle component images that

did not get used in the second-stage process. Each of these images was compressed

spacially into a form usable for transmission. The compression was a JPEG 2000

standard, which involved converting the principle component images, in a ‘.raw’ format

to a ‘.jp2’ format. To do this, each principle component image was loaded into JASC

Paint Shop Pro. In the conversion, the number of bytes used to store the contents of the

image (file size) is reduced. The lower the file size converted to, the greater the loss of

information from the reconstructed images. However, the tradeoff to converting to a

small file size is having a larger capacity for transmission other sets of image data. With

limited transmission bandwidth, increasing the compression with minimal loss is

important.

23

This thesis is not interested so much in the effect of the degree of compression of

an image. The concern was what effect the choice in file number per KLT group, during

the first-stage processing, had on image degradation after reconstructed. We therefore

used a fixed reference compression rate for this paper. This was done by choosing a

reasonable effective coding bit rate, R, of 0.25 bits per pixel.

R is essentially the number of bits used to code a pixel (in this case 8 bits) divided

by a compression factor, CR. The compression factor is the total number of pixels in a set

of files compressed together divided by the desired number of bytes used to compress

these files [6]. The pixels in a set, in our case, are the total number of pixels in all the

files within a set of principle component images that underwent a particular KLT

operation. Therefore second-stage KLT files were placed in a different set than the

utilized first-stage KLT files.

The total number of bytes allocated for the compression was also determined in a

particular way. It was not only made up of the sum of the compressed file sizes for every

principle component image in a set, but also an overhead file providing all the parameter

information used in the particular KLT process. The overhead file had to be transmitted

too, because it was necessary for the reconstruction of the images. The left over available

bits were distributed among the principle component images in a set.

The distribution of bits was not even, however. It turns out that the degree of

compression that can be performed on each file of a set, and maintain a fairly consistent

degree of degradation, can be set proportional to the square root of its eigenvalue (or

standard deviation) [6]. Therefore, the compressed-file sizes chosen for each principle

component was determined under two conditions. Firstly, a file size was dictated by the

24

images relative standard deviation. Secondly, the sum of the bytes used for the file sizes

equaled the total number of bytes available to allow for an effective coding bit rate of

0.25 bpp.

4.3 Reconstruction

The reconstruction process began by spacially decompressing the ‘.jp2’-converted

images back into their ‘.raw’ form. The following steps were essentially the reverse of the

spectral compression protocol used. An Inverse KLT (IKLT) was performed on the set of

decompressed second-stage KLT principle component images. Again, each set was

derived from the grouping of all first, second, third, and fourth principle component

images from the first-stage KLT. Once these principle components were reconstructed,

they were reordered with the leftover decompressed first-stage KLT principle component

images. At this point, a final IKLT was performed on each of the groups of principle

component images to yield reconstructed spectral component images.

4.4 Classification

Once spectral images of a scene are obtained, whether directly or through a

transmission and reconstruction scheme, the images must be used to provide useful

information. The spectral signature of locations in a scene can be used to identify the

types of ground cover. There are many methods for such classification, each with its own

advantages and disadvantages.

An unsupervised classification approach was applied to the original and

reconstructed spectral images. The scene of San Francisco was resolved into 6 distinct

classes. To begin the classification process, 6 initial spatial locations were chosen by

referencing particular spatial coordinates for 6 distinct features in previous performed

25

classification on the same spectral images [9]. These same ‘centroids’, each marked by a

unique set of reflection intensities, were used as the initial starting points for the

classification of all the spectral image sets obtained. All other pixels in the scene were

assigned to the class associated with a centroid. This assignment was based on a

minimum Euclidean distance critereon [10]. Pixels were assigned to the class of the

centroid that they were the minimum, spectral-vector distance from. For the next

classification iteration, a new centroid was determined for each class by taking the

average of the intensities of each pixel in a class. The pixels were then reclassified in the

same manner. The process of reclassifying pixels continued until the sum of squared

distances from a centroid to its associated class pixels changed relatively little at the next

iteration (see Appendix for code). A root-mean-squared-error (rmse) criterion was used

such that when an iteration caused less that a 1.8% change in pixel classification, the

process would stop (described in section 4.5.1). In all cases, this occurred in less than 10

iterations. The result of this ‘general hill climbing’ approach was a ‘thematic map’

distinguishing different classes of objects in the scene by different colors.

4.5 Quantitative Assessment of Reconstruction

There have been a number of methods proposed for achieving a suitable measure

of image degradation of a decompressed image. A popular theoretical measure is to

simply calculate the total root-mean-squared error between the original and reconstructed

spectral images. However, for the practical application of ground-cover identification, it

is also important to compare error in the classification process via confusion matrices. In

this thesis, both were investigated.

26

4.5.1 Root-mean-squared Error

To measure the degree of degradation of the reconstructed images, each resulting

spectral image was compared to its respective original spectral image, pixel by pixel.

This was performed for all three types of groupings of spectral images: 20, 16, and 8 files

per group. The effectiveness of each of the three types of grouping was compared by

looking at their overall percent root-mean-squared error (rmse), with respect to their

corresponding original spectral images. The calculations were done using code written in

Matlab to speed up the processing (see Appendix). Firstly, the sum of the squared

differences between each original and reconstructed pixel pair, for all spectral images of a

particular grouping, was determined. The cumulative summing of squared differences

was performed over the entire spectrum. This yielded the total error square, which was

divided by the total number of pixel pairs (the number of pixels in an image, 256x256,

times the number of images used, 144). This last calculation gave the mean square error.

Taking the square root of this value produces the root-mean-square error (rmse). Finally,

by dividing the rmse by 255 (the largest value that a pixel can be) the error becomes

scaled. Multiplying the result by 100 gives the percent rmse. We looked at the percent

rmse for each of the three file size groupings and compared them.

4.5.2 Confusion Matrix

A confusion matrix reveals the percentage of false positives and false negatives in

the classification process described earlier. The pixel classification received from the

thematic map, obtained using the original spectral images, was treated as the true

classification of the scene. The pixel classifications for each reconstruction scenario was

compared to each pixel or the true classification. The percentage of correctly identified

27

pixels in a class (relative to the reconstructed classification) is placed in the right most

column in the confusion matrix (Tables 5-1, 5-2, 5-3). These values are the ‘User

Accuracy.’ The bottom row of the matrix contains the percentage of correctly classified

pixels (relative to the true classification). These values are the ‘Producer Accuracy.’

Finally, the total percentage of correctly identified pixels can be found in the lower right

corner of the matrix.

4.6 PC Contribution to Second-stage KLT

In an effort to investigate how much of the first-stage principle component image

information is decorrelated in the second-stage KLT operation, the characteristics of the

second-stage transformation were looked at. The value of each of the elements in an

eigenvector indicates the relative amount of information that each of the first-stage

principle component images, used in the transformation, contributes to its single

corresponding principle component image received at the second stage. After the first-

stage KLT, all of the PCs of the same level were combined (i.e. PC 1 from each group,

where 1 is the level number and i value in Figure 4-1). Again, each PC image represented

the transformation of a particular grouping of spectral images. An entire set of

eigenvectors results for a single equi-level PC, second-stage transformation.

One noted phenomena was that, given a particular equi-level PC transformation,

each and every eigenvector, of a single transformation, could all contain a 0 (accurate to

2 significant digits, in this thesis) as their value at a particular component index. This

indicated that the first-stage PC image associated with the index value did nothing to

contribute to the second-stage decorrelation. In other words, to begin with, information

from this first-stage PC image was not correlated with the other first-stage PC images in

28

its KLT group. This is equivalent to not including the uncorrelated first stage PC image in

the calculation of the second stage transformation matrix. Under such scenarios the

percentage of contributing first-stage PC images in a transformation group was compared

at each PC level ( i value in Figure 4-1) and across the grouping scenarios of 8, 16, and

20 files/group. Percentages were calculated by dividing the number of eigenvalue indices

that did not contain the value 0 across all eigenvectors by the total possible number of

eigenvalue components.

29

Chapter 5 Results

When the percent rmse values between original and reconstructed images were

determined, a general trend between file number per spectral grouping and image

degradation was observed. At a coding bit rate of 0.25 bpp, as the number of files per

grouping was decreased, the overall percent rmse decreased (Figure 5-1).

Figure 5-1 Reducing the number of spectral images used per group in 1st stage KLT operations had a general effect of reducing the degradation of the reconstructed image.

These quantitative results were consistent with the qualitative results as well. By

inspection of the thematic maps obtained from the classification process, one can see a

trend in image resolution and accuracy. The thematic map constructed from the original

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

1

Percent RMSE (%

)

20 images/group 16 images/group 8 images/group

Spectral-grouping Effect on Image Degradation

30

set of spectral images showed very distinct features of the scene of San Francisco (Figure

5-2). Comparing this to the thematic map generated from the reconstructed spectral

Figure 5-2 Ground Truth Thematic Map. The thematic map constructed from the original spectral images shows distinct features of San Francisco. images of the 8 file/group compression scheme showed some minor changes in the shape

of the class features (Figure 5-3). However, major trends in feature shape were not lost.

The 16 file/group thematic map had much more obvious deviations from the original

31

Figure 5-3 8 File/group Thematic Map. The thematic map constructed from the 8 file/group spectral images shows minor losses in feature shapes compared to the original. thematic map (Figure 5-4). Much of the resolution of the features marked in yellow and

teal had degraded. In addition to poor resolution, one could see loss in the major contours

of the features for a 20 file/group scenario (Figure 5-5).

32

Figure 5-4 16 File/group Thematic Map. The thematic map constructed from the 16 file/group spectral images shows some evident resolution loss of features compared to the original.

33

Figure 5-5 20 File/group Thematic Map. The thematic map constructed from the 20 file/group spectral images shows some evident contour loss of features compared to the original. Looking more closely at the classification results via the associated confusion

matrix, showed a minor trend in User and Producer Accuracy. A more evident trend was

found in the overall accuracy. The likelihood of getting classification false positives and

false negatives appeared to increase a bit in general as more files/group are used in the

compression process (Tables 5-1, 5-2, 5-3). But the number of correctly identified pixels

did not seem to necessarily increase for a given class as the number of files/group

34

decreases. However, in general, the number of total correctly identified pixels seemed

more likely to increase with the decrease in file/group number.

Class 1 2 3 4 5 6

Row

Sum

User

Accuracy

(%)

1 6026 0 0 0 23 116 6165 97.75

2 20480 24 23 581 0 21108 97.02

3 0 20 3885 89 0 0 3994 97.27

4 0 844 13 10176 0 0 11033 92.23

5 74 70 0 0 9776 12 9932 98.43

6 116 0 0 0 5 13183 13304 99.09

Col. Sum 6216 21414 3922 10288 10385 13311 65536

Producer

Accuracy

(%) 96.94 95.64 99.06 97.99 94.14 99.04 96.93

Table 5-1 8 File/group Confusion Matrix. The confusion matrix for a thematic map comparison of an 8 file/group compression to the original spectral images. One can see the degree of false positives and false negatives for each class.

Class 1 2 3 4 5 6

Row

Sum

User

Accuracy

(%)

1 6117 0 0 0 224 44 6385 95.80

2 0 19555 2 2610 0 0 22167 88.22

3 0 150 3468 36 0 0 3654 94.91

4 0 0 452 7642 0 0 8094 94.42

5 0 1709 0 0 10138 0 11847 85.57

6 99 0 0 0 23 13267 13389 99.09

Col. Sum 6216 21414 3922 10288 10385 13311 65536

Producer

Accuracy

(%) 98.41 91.32 88.42 73.59 97.62 99.67 91.84

Table 5-2 16 File/group Confusion Matrix. The confusion matrix for a thematic map comparison of a 16 file/group compression to the original spectral images. One can see the degree of false positives and false negatives for each class.

35

Class 1 2 3 4 5 6

Row

Sum

User

Accuracy

(%)

1 5923 0 0 0 284 545 6752 87.72

2 0 18580 171 1235 1110 0 21096 88.07

3 0 60 3673 102 0 0 3835 95.78

4 0 1521 78 8950 0 0 10549 84.84

5 152 1253 0 1 8967 78 10451 85.80

6 141 0 0 0 24 12688 12853 98.72

Col. Sum 6216 21414 3922 10288 10385 13311 65536

Producer

Accuracy

(%) 95.29 86.77 93.65 86.18 86.35 95.32 89.69

Table 5-3 20 File/group Confusion Matrix. The confusion matrix for a thematic map comparison of a 20 file/group compression to the original spectral images. One can see the degree of false positives and false negatives for each class. The results suggested that one can optimize image reconstruction and feature

identification through a change in file number per group. Additionally, second-stage KLT

results suggested an optimization in number and choice of first-stage PC images for inter-

group decorrelation. Second-stage KLT was performed on the collection of the first 4 PC

images of the first-stage. Inspection of the eigenvector components of the second-stage

KLT showed situations in which no significant PC image decorrelation occurred for

particular PC groupings after the first stage. The percentage of PC images that were used

in the second-staged KLT were noted for each compression scheme. Results showed that,

in every case of choice in file number per group, fewer first-stage PC images contributed

to second-stage decorrelation (Figure 5-6). Furthermore, over a 4-PC-image-deep second-

stage transformation, the percentage of first-stage PC image contribution dropped off

faster the fewer the number of spectral images that were used per group. The actual PC

images that did not contribute were also identifiable from the set of eigenvectors by

noting the index value of the eigenvectors, which is associated with PC level.

36

Figure 5-6 Second-stage Decorrelation Analysis. The percentage of combined equi-level PC images, from the first-stage KLT, that contribute in the second stage KLTs, decreases with the depth of PC level used. Furthermore, the fewer spectral images used per group, the faster the drop-off in contributing first-stage PC images with PC depth.

37

Chapter 6 Conclusion and Future Work

The proposed 2-stage KLT method addresses a number of practical limitations in

transmitting a large collection of spectral images. For instance, in some cases when a

large spectrum of hyperspectral images must be captured at once, multiple independent

terminals must be used. Furthermore, the processing may also have to be done through

separate terminals. Therefore, the compression technique has to be designed to perform

many local transformations without misrepresenting the areas of spectral correlation. At

the same time, with many images to be processed, it is more practical to rely on a method

that can treat the spectral images as groups of correlated images rather than individually.

However, the grouping should be as methodical as possible, to maximize the accuracy of

the reconstructed image, while still keeping the protocol simple.

The compression approach used in this project acknowledges the limitations of

processing a large set of images simultaneously. While there are other simpler techniques

that perform the transformation, like the bidirectional interband prediction method, they

are not always so straightforward and can require too much specialized treatment to each

band [3]. This is not always worthwhile when dealing with a large number of bands. The

application of the local KLT would therefore, more appropriate for a study like this one.

However, even though the local KLT finds an optimal transformation matrix for a single

spectral partition, it requires a single transformation operation of all the rest of the

spectral images at once. This operation can be computationally inefficient, especially if

the number of other spectral images is very large. The two-stage KLT gets around this by

partitioning the spectral images into more than two partitions, making the transformation

operations less computationally demanding. The local KLT may actually produce better

38

reconstruction than the two-stage KLT because it focuses on finding optimum

transformation parameters to produce minimum error. But this computational approach

seems to be a much larger price to pay than the proposed method. The local KLT does

not do much to elevate the biggest criticism of the KLT algorithm – the computational

burden. The distributed KLT makes even more dramatic computational efforts into

further optimizing reconstruction [5]. This can be highly impractical when the most

critical factor is addressing interband correlation, which the two-stage KLT does.

Recombining groups of independent principle component images at many KLT

stages would prove very effective and would ensure benefits from all correlations

throughout the spectra [8]. The computation is also much more straightforward that the

distributed KLT. However, if the number of files per group is small and/or the number of

groups is large, the processing power necessary to address the less significant correlations

at the later stages, may not make up for the benefits. This is precisely what the sub-

optimal KLT fails to address. A two-stage KLT, may be enough to capture the significant

spectral correlations. Like the sub-optimal KLT, the two-stage method used an arbitrary

number of files per spectral grouping. However, unlike the sub-optimal KLT, this paper

further provides some rational on the side effects and tradeoffs with a choice in file

number per group. Such information can be used to optimize file number per group

choice.

In general, it appears that decreasing the number of spectral images used for the

first-stage local KLTs yields overall less image degradation and classification

inaccuracies. This also means more KLT operations at the first stage. However, to get full

decorrelation between groups, fewer PC images, and possibly fewer PC depth levels, are

39

required at the second-stage. The sub-optimal KLT that was described uses a uniform PC

depth at every KLT stage. However, depending on the contribution distribution of PC

images to later KLT stages, the contribution of a fewer number of images or PC levels

may be necessary to yield comparable decorrelation results. In other cases, the number of

PC images and levels needed to completely decorrelate images at later stages, may be

insufficient with an arbitrary suboptimal KLT file per group number. One can provide

this rational to optimize the depth of decorrelated second-stage PCs on the data generated

from this paper, while using the scenario yielding the best reconstruction and

classification.

For the experiment described in this thesis, one would want to use 8 files/group,

but will require at least 4 levels deep in second-stage PC decorrelation to get at least

roughly 10% first-stage PC contribution to the second stage. The reason for this is that at

the fourth PC level already roughly only 10% of the first-stage PC images contribute

(Figure 5-6). In fact, from the inspection of the eigenvectors one can actually select the

particular PC images needed for the second KLT. Again, this is all determined by looking

at the values of the eigenvectors of the second-stage transformation. 10% at the fourth PC

level means that one tenth of the collected PC 4 images from the first-stage KLT were

correlated with the rest of the PC 4 images. In other words, across all the eigenvectors for

the particular second-stage transformation, one-tenth of the eigenvector components

contained non-zero values. Therefore, one can get the same results by simply removing

the PC 4 images whose corresponding eigenvector indices are 0, and then perform the

transformation again. In this way computational efforts can be saved.

40

The results of my experiments indicated at the smallest file/group number, 8, only

needed PC 1 through PC 4 to almost completely decorrelate the images. In fact, was this

choice in file/group number and PC depth of second-stage compression that allowed the

quantitative goals, described in the introduction, to be met. Again, one quantitative goal

was to compress the 144, 256x256 pixel spectral images at 0.25 bpp such that the

reconstructed images exhibited a root-mean-squared error of less than 1%. With 8

files/group a root-mean-squared error of 0.29 %. The second quantitative goal was to

receive a total classification accuracy of greater than 95%. A 6-class, unsupervised

minimum Euclidean distance, general hill climb classification scheme was considered.

Iterations were set to stop after less than a 1.8% change in pixel classification was

achieved. Again, this second goal was achieved by using 8 files/group to yield a total

classification accuracy of 97.93%. Therefore all the quantitative goals of this thesis were

met.

However, this does not suggest that, in every compression scenario, optimal

reconstruction will occur with 8 files/group and a PC depth of 4. Results may vary with

choice in image, the resolution of the spectral imager used, as well as changes in test

parameters (i.e. a use of 0.25 bpp and 1.8% pixel change in the general hill climb

method). My results have merely suggested that minimizing file number per group seems

to yield more accurate reconstruction results. My results also indicate that the PC depth

used at second-stage KLT is a function of the file number per group used and that this

depth is determinable by inspection.

To explain the apparent relationship between file/group number and depth of

second-stage PC image decorrelation, a hypothesis is proposed. It appears that the fewer

41

the number of images in a group used at the first stage, the more accurate the linear

approximation of the KLT becomes. Assuming a linear relationship between spectral

dynamics is a bold statement to make, but with fewer data samples simpler models, such

as a linear relationship, may be fairly accurate. This could explain the overall lowest rmse

results with 8 files/group.

However, one can expect that within a very narrow spectral range, few dynamics

are present. But over a larger range, such as a spectral group containing 20 files, many

major spectral dynamic relationships may be present. Therefore the probability of any

two spectral groups of 20 images sharing the same dynamics at a given PC level is higher

than say groupings of 8 spectral images. This could explain why with the higher

files/group number much more second-stage decorrelation at lower PC depths was taking

place. Again, at a PC level of 4 almost all images were uncorrelated when using 8

files/group. This was probably because not much dynamic information could be found at

this level over such a small spectral range. However, with 16 files/group and 20

files/group, much more decorrelation was possible beyond a PC depth of 4 at the second

stage. Exploiting this apparent relationship between file/group number and required PC

level decorrelation at the second stage of KLT operations is what makes this method so

advantageous.

Although, the two-stage KLT method, introduced here, has some advantages over

existing compression techniques, it does have some of its own limitations as well. The

major drawback on this compression methodology stems from the fact that at the second

stage KLT only PC images from the same depth level are combined. This falsely makes

the assumption that all the dynamics captured at one PC level can only be contained in

42

the same PC level for other spectral group ranges. The distributed-KLT method takes this

into account by including the contribution of every set of transformed images to every

other set of transformed spectral images.

In addition to a lack of correlation optimization, the described compression

scheme does not propose an optimal file per group number to be used. The number of

files chosen per group was arbitrary. It simply depends on the number of files one can

afford to process at each stage and how accurate a reconstruction one wishes to get. For

the most part, it was noticed that reducing the number of files used for the first-stage

KLT acted to improve the reconstruction results overall. However, as was observed, this

is not true at the local spectral range level or the local class assignment level. In

conclusion, the 2-stage KLT method is useful when a direct KLT on all spectral images is

either impractical or not possible. It caters itself to scenarios were the number of KLT

operations must also be minimized but where exploiting interband correlation is

important. One can use the trade offs, shown, among number of KLT operations, number

of files undergoing a KLT operation, and desired reconstructed image accuracy, to

determine one’s ideal layout for his/her 2-stage KLT approach. As was shown, it is

possible to achieve less than 1% rmse and greater than 95% total classification accuracy.

Future work could look deeper into the spectral grouping preferences in particular

spectral ranges. This might develop into a methodology that could be used in determining

an optimal set of file-number groups to be used for any set of spectral images. One could

further write an efficient algorithm that compares mean squared errors at the image level

instead of the group level, in order to further reduce the total percent rmse. It is also

worth trying a different KLT algorithm. An algorithm that makes use of the factorization

43

method may prove to be more accurate at determining transformation parameters [7]. An

investigation of the spacial compression algorithm could also be done. In particular, one

could compare the use of JPEG2000 in this paper to that of simply applying the recently

introduced JP3D, the 3D extension of JPEG2000.

JP3D is the new standard for spacial compression of 3D data. It has particularly

useful applications in such fields as medical imaging, where storage of large volumetric

data sets is necessary. The JP3D specifications essentially do not introduce new coding

concepts but simply extend those of JPEG2000 parts 1 & 2 (embedded block coding by

optimized truncation (EBCOT) based on the principles of layered zero coding (LZC)) to

a 3rd dimension [11]. The JP3D codec first partitions the volumetric data set (typically a

stack of 2D images) into tiles of volumetric blocks that are independently encoded as

separate grayscale datasets. All the tile sample coefficients are then filtered via a discrete

wavelet transform, using the Mallat decomposition pattern [11]. From here, the codec

partitions the resulting sub-bands into dyadically-sized code-blocks, independently

encoded by the EBCOT coder (Figure 6-1). The code-block coefficients are processed

44

Figure 6-1 Partitioning and Encoding in JP3D. After a 3D data set is subdivided, each subdivision is broken down into code-blocks, which are encoded independently [11]. four adjacent bits at a time within a bit plane, before moving down to the next layer

(Figure 6-2). The bit plane pattern is identical to that used in JPEG2000 part 1 [11].

Figure 6-2 JP3D Scanning Pattern. 3D datasets are processed along bit planes and from one slice to the next [11].

45

The JP3D compression scheme could also be applied to spectral images used in

remote sensing, where the 3rd axis could simply be replaced by the spectral information

of a 2D scene. The algorithm does have a conceivable application advantages over the 2-

stage KLT. Because JP3D uses a wavelet transform approach in all three dimensions, as

opposed to a linear transform, such as the KLT, it is more suitable for transmitting data a

few lines at a time [11]. Our approach does not have this kind of scan-based-mode

flexibility over the spectral dimension. It requires the collection of statistics over the

whole image and selects spectral ranges before transmission can take place.

However, as a hyperspectral imaging compression algorithm, JP3D seems to have

some computational disadvantages and could possibly be undesirable as a compression

model compared to the proposed approach. JP3D’s process of partitioning, sub-

partitioning, and then filtering at the tile sample coefficient level, can be very involved

[11]. If spectral linearity can be assumed, computation can be greatly, and sufficiently,

simplified. This simplification is the aim of the 2-stage KLT approach. However,

adequate, full spectrum, yet intelligent, decorrelation is also the focus of the proposed

approach. JP3D makes efforts to perform decorrelation within and between every bit

plane [11]. However, the information provided by the 2-stage KLT justifies the

needlessness of attempting decorrelation between select spectral planes from different

spectral sets. Therefore, unlike JP3D, the 2-stage KLT avoids unnecessary efforts to

further compress an image. It should also be pointed out that, just in terms of transform

models, it has been shown that using a KLT in the spectral dimension, followed by a

JPEG 2000 discrete wavelet transform (DWT) in the spacial dimension, yields superior

rate distortion performance than a JPEG 2000-based DWT in all 3 dimensions [12]. This

46

suggests that the 2-stage KLT, which makes use of KLT and the JPEG 2000 DWT, may

be, in general, a more desirable model. It has also been proposed that application of JP3D

may be ineffective at fully addressing issues specific to hyperspectal imagery [13].

47

BIBLIOGRAPHY

[1] Saghri, John. “Principle Component Analysis: Lecture Materials 10.” California Polytechnic State University, San Luis Obispo. Spring 2008.

[2] Penna, B. “Transform Coding Techniques for Lossy Hyperspectral Data

Compression.” IEEE Transaction on GeoScience and Remote Sensing Vol. 34, NO. 5, May 2007.

[3] Ashok, Rao. “Multispectral Data compression Using Bidirectional Interband

Prediction.” IEEE Transactions of Geoscience and Remote Sensing Vol. 34, NO. 2, March 1996.

[4] Shlens, Jonathon. “A Tutorial on Principle Component Analysis.” Systems

Neurobiology Laboratory, Stalk Institute of Biological Studies. Version 2, Dec. 2005.

[5] Gastpar, Michail. “The Distributed Karhunen-Loeve Transform.” IEEE

Transactions on Inforamation Theory Vol. 52, No. 12, Dec. 2006. [6] Saghri, John. “Laboratory Experiment 5: Bandwidth Compression of

Multispectral Imagery Data.” California Polytechnic State University, San Luis Obispo. June 2008.

[7] Press, William. Numerical Recipes: The Art of Scientific Computing. New

Rochelle: Cambridge Press, 1988. [8] Wongsawat, Yodchanan. “Integer Sub-optimal Karhunen-Loeve Transform for

Multi-channel Lossless EEG Compression.” University of Texas at Arlington. 2004.

[9] Saghri, J. , Tescher, A. , Planinac, A. “Band Compression of Hyperspectral

Imagery Data using a Simplified KLT/JPEG 2000 Approach.” Proceedings of SPIE, vol. 558, no. 68, August 6, 2004.

[10] Saghri, John. “Classification: Lecture Materials 5.” California Polytechnic State

University, San Luis Obispo. Spring 2008. [11] Bruylants, Tim. “Volumetric Image Compression with JPEG 2000.” SPIE. 2007.

[12] Du, Q. “Hyperspectral Image Compression Using JPEG2000 and Principal

Component Analysis.” IEEE Geoscience and Remote Sensing Letters 4(2), 201-205, April 2007.

[13] Pickering, M. “An Architecture for the Compression of Hyperspectral Imagery.”

In G. Motta, F. Rizzo, and J.A. Storer (Eds.). Hyperspectral Image Compression. Chapter 1, pp. 1-34. New York: Springer, 2006.

48

APPENDIX

The Karhunen-Loeve Transformation

Concept

Suppose we have a picture of a scene and the picture is separated into its spectral

images. Each spectral image contains pixel values indicating the intensities of the

ground-cover reflections of particular a spectral band (i.e. the Visible Red wavelength

content in the scene). Each spectral image can also be thought of as images taken of an

event by a single camera from a particular reference angle in space. Let’s use an example

to explain this [4].

Assume the scene, or event, that we are observing is a ball on a spring, moving

back and forth along one dimension in space. We observed the position of the ball in

space over time using cameras taken from various angles, relative to the direction of

motion of the ball (Figure A-1). The assumption that the KLT method makes is that the

Figure A-1 KLT Conceptual Example. Spectral images of a scene can be thought of as taking pictures of an event, such as a moving ball on a spring, where each image is taken from a different perspective [4].

observations taken by the camera, in this case, position relative to the 2D plane of the

camera lens, is a movement with displacement in one linear dimension, but within a

multiple dimensional space. Each camera can only detect the components of the x, y,

directions of movement of the ball in space, relative to the positioning of it 2D lens.

Further, each camera has the chance that it can detect data (movement) along any

combination of the fixed, reference dimensions x, y,

perchance be positioned so that it

detect movement of the ball, moving in the x direction. If the ball is moving along only

one dimension than we need at least one camera to fully observe it, but facing a

along the direction of movement.

49

1 KLT Conceptual Example. Spectral images of a scene can be thought of as taking pictures of an event, such as a moving ball on a spring, where each image is taken from a different perspective [4].



multiple dimensional space. Each camera can only detect the components of the x, y,



combination of the fixed, reference dimensions x, y, and z. A camera’s lens could

chance be positioned so that it is perpendicular to the x axis. It therefore would not


one dimension than we need at least one camera to fully observe it, but facing a

along the direction of movement.

1 KLT Conceptual Example. Spectral images of a scene can be thought of as taking pictures of an event, such as a moving ball on a spring, where each image



multiple dimensional space. Each camera can only detect the components of the x, y, z



and z. A camera’s lens could

is perpendicular to the x axis. It therefore would not


one dimension than we need at least one camera to fully observe it, but facing a plane

50

Now suppose the scene contains three simultaneous events, where each event is a

ball moving along one of the three orthogonal directions. If we want to fully describe the

events of three balls moving in mutually orthogonal directions than we need at least three

cameras, all of which are orthogonal to each other and provide unrelated data. One,

however do not know how many dimensions we are able to detect from each camera, as

this is dependent on the direction from which the camera is observing the events.

However, if the mapped data of a camera moves with high correlation to the mapped data

of another camera, then both cameras are most likely detecting the same event of

movement along one dimension. So if two cameras are being used to capture the same

event and produce separate images, one can simply combine the two images into one and

get rid of any redundant movement information. This is exactly what the KLT operation

does. It takes highly correlated spectral image information and combines it, effectively

reducing the number of images need to describe an event. But if the scene contains

multiple independent events, which are changing along different, orthogonal dimensions,

one would need to separate all the compressed camera image data into new images that

each capture a different, uncorrelated event. In the case of three orthogonal-moving balls,

we would combine and partition the data collected by multiple cameras into three

principle component images, each describing a different ball’s, one-dimensional motion.

Now that we have a better idea of how the KLT operation works, we can tie our

example into the application of ground-scene image compression. Think of a single

camera recording in time, from our example, as being homologous to a single spectral

image pixel value. A single camera captures many positions of multiple balls in the

scene. Likewise, a single spectral image band captures many pixel intensities of multiple

51

reflection events from ground features. Therefore, choosing a particular point in time

whose camera image data you want to look at is analogous to choosing a particular pixel

whose spectral value you want to look at.

Derivation

The first step in converting a series of spectral images into a set of principle

component images, via the KLT method, is to determine how related spectral images are.

This is done by finding the covariance between every combination of spectral values for

all the pixels in the image [4]. We begin by defining a matrix, X. Every column in this

matrix consists of the reflection intensities of each spectral image for a particular pixel.

There are n rows for each of the n pixels in the scene image. We can define the

covariance matrix as:

CX = (1/(n-1))XXT .

Our goal is to convert every spectral image into an equal set of principle component

images. Essentially, we are recombining the spectral information into images of

uncorrelated data, each of which describes a different, independent spectral trend in the

scene. However, not every principle component image is capturing a very significant

trend. The new derived principle component images will be defined as the matrix, Y,

structured in the same way as X. We assume that any spectral image pixel value may be

linearly related to any other spectral image value, such that the spectral measurements are

orthogonal to one another. If we think of the spectral values as measurements along

orthogonal axes of trends in the scene, we can rotate the axes such that a set of linearly

correlated spectral data will produce the largest variance along its new axis (Figure A-2).

Figure A-2 Effect of Spectral Axis Rotation on Data Variance. If we assume that the measurements of the spectral bands are coordinates of a linear relationship between sets of bands, then the spectral band axes can be rotated. The closer one rotates an axis toward a data trend, the larger the variance along the axis becomes. Likewise, the accuracy of the measurement of the trend will increase, therefore the signalnoise ratio will increase [4]. The movement of data of a particular trend is expected to move along this particular new

axis. Therefore, the further one’s axis of measurement is rotated from this principle axis,

the more inaccurate (more noisy) the measurement of the trend’s signal alon

becomes. The set of values along the new, rotated axes are the set of values provided in

each new principle component image. This axis rotation process and subsequent

transformation of pixel values can be expressed as:

Y = ΡΡΡΡX ,

where the rows of ΡΡΡΡ are the

to a new set of values.

If we were to have our new principle component images be uncorrelated with

each other, and each provide an independent set of trends in the scene, then the

52

2 Effect of Spectral Axis Rotation on Data Variance. If we assume that the measurements of the spectral bands are coordinates of a linear relationship between

s of bands, then the spectral band axes can be rotated. The closer one rotates an axis toward a data trend, the larger the variance along the axis becomes. Likewise, the accuracy of the measurement of the trend will increase, therefore the signal

ratio will increase [4].

The movement of data of a particular trend is expected to move along this particular new


the more inaccurate (more noisy) the measurement of the trend’s signal alon



transformation of pixel values can be expressed as:

are the principle components of X that map the pixel spectral values



2 Effect of Spectral Axis Rotation on Data Variance. If we assume that the measurements of the spectral bands are coordinates of a linear relationship between

s of bands, then the spectral band axes can be rotated. The closer one rotates an axis toward a data trend, the larger the variance along the axis becomes. Likewise, the accuracy of the measurement of the trend will increase, therefore the signal-to-

The movement of data of a particular trend is expected to move along this particular new


the more inaccurate (more noisy) the measurement of the trend’s signal along the axis



that map the pixel spectral values



53

covariances between the data, mapped to the principle component axes, should be zero.

At the same time, the new data mapped to each particular principle component should

contain a non-zero variance. Therefore, the goal of the KLT operation is to provide a

covariance matrix for Y whose off-diagonal values are zero and whose diagonal values

are non-zero. In theory, this can be done and we show it through some basic principles in

linear algebra.

Firstly, as with X, we can define the covariance matrix for Y as:

CY = (1/(n-1))YYT .

By the identity provided earlier, we can rewrite this equation and come to the following

conclusion:

CY = (1/(n-1))(ΡΡΡΡX)(ΡΡΡΡX)T

= (1/(n-1))PXXTΡΡΡΡ

T

= (1/(n-1))ΡΡΡΡ(XXT)ΡΡΡΡT

= (1/(n-1))ΡΡΡΡCXΡΡΡΡT .

Because CX is a symmetric matrix, it can be diagonalized by an orthogonal matrix of its

eigenvectors:

CX = EDET ,

where E is the matrix of eigenvectors arranged by columns and D is a diagonal matrix.

We also note that the inverse of the orthogonal matrix, ΡΡΡΡ, is its transpose. With these

observations in mind, we can further rewrite the equation for CY:

CY = (1/(n-1))ΡΡΡΡCXΡΡΡΡT

= (1/(n-1))ΡΡΡΡ(ΡΡΡΡTDΡΡΡΡ)ΡΡΡΡT

= (1/(n-1))(ΡΡΡΡΡΡΡΡT)D(ΡΡΡΡΡΡΡΡT)

54

= (1/(n-1))(ΡΡΡΡΡΡΡΡ-1)D(ΡΡΡΡΡΡΡΡ-1)

CY = (1/(n-1))D .

It is clear from this reduction that in order to reach our goal of diagonalizing CY we must

find the appropriate choice of ΡΡΡΡ. As one will noticed from the way one comes up with this

final equation, the rows of ΡΡΡΡ will be the eigenvectors. The associated eigenvalues will be

the diagonal values of CY.

For every eigenvector and associated eigenvalue we have a respective principle

component image. The larger the eigenvalue in a set, the more information content that

its respective principle component image contains of the scene. However, 95% of the

information content of all the spectral data resides in the first two or three spectral

component images [1]. Therefore, after such a compression method, often only the first

three principle component images are needed to reconstruct the original spectral images

without significant information loss.

55

Matlab Code

% Calculation of percentage root mean squared errors for each of the % spectral images (simplified for debugging) o1=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M97.raw',256,256); o2=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M98.raw',256,256); o3=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M99.raw',256,256); o4=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M100.raw',256,256); o5=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M101.raw',256,256); o6=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M102.raw',256,256); o7=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M103.raw',256,256); o8=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M104.raw',256,256); o9=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M121.raw',256,256); o10=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M122.raw',256,256

); o11=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M123.raw',256,256






); %o17=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M17.raw',256,25

6); %o18=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M18.raw',256,25



6); r1=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M97R.raw',256,256); r2=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M98R.raw',256,256); r3=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M99R.raw',256,256); r4=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M100R.raw',256,256

); r5=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M101R.raw',256,256

); r6=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M102R.raw',256,256

);

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r7=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M103R.raw',256,256);










%r17=ReadRawImage('C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M17R.raw',256,256);




rmse1_matrx=cat(3,o1,r1,zeros(256,256)); rmse2_matrx=cat(3,o2,r2,zeros(256,256)); rmse3_matrx=cat(3,o3,r3,zeros(256,256)); rmse4_matrx=cat(3,o4,r4,zeros(256,256)); rmse5_matrx=cat(3,o5,r5,zeros(256,256)); rmse6_matrx=cat(3,o6,r6,zeros(256,256)); rmse7_matrx=cat(3,o7,r7,zeros(256,256)); rmse8_matrx=cat(3,o8,r8,zeros(256,256)); rmse9_matrx=cat(3,o9,r9,zeros(256,256)); rmse10_matrx=cat(3,o10,r10,zeros(256,256)); rmse11_matrx=cat(3,o11,r11,zeros(256,256)); rmse12_matrx=cat(3,o12,r12,zeros(256,256)); rmse13_matrx=cat(3,o13,r13,zeros(256,256)); rmse14_matrx=cat(3,o14,r14,zeros(256,256)); rmse15_matrx=cat(3,o15,r15,zeros(256,256)); rmse16_matrx=cat(3,o16,r16,zeros(256,256)); %rmse17_matrx=cat(3,o17,r17,zeros(256,256));

57

%rmse18_matrx=cat(3,o18,r18,zeros(256,256)); %rmse19_matrx=cat(3,o19,r19,zeros(256,256)); %rmse20_matrx=cat(3,o20,r20,zeros(256,256)); for i=1:256 for j=1:256 rmse1_matrx(i,j,3) = (rmse1_matrx(i,j,2) - rmse1_matrx(i,j,1))^2; rmse2_matrx(i,j,3) = (rmse2_matrx(i,j,2) - rmse2_matrx(i,j,1))^2; rmse3_matrx(i,j,3) = (rmse3_matrx(i,j,2) - rmse3_matrx(i,j,1))^2; rmse4_matrx(i,j,3) = (rmse4_matrx(i,j,2) - rmse4_matrx(i,j,1))^2; rmse5_matrx(i,j,3) = (rmse5_matrx(i,j,2) - rmse5_matrx(i,j,1))^2; rmse6_matrx(i,j,3) = (rmse6_matrx(i,j,2) - rmse6_matrx(i,j,1))^2; rmse7_matrx(i,j,3) = (rmse7_matrx(i,j,2) - rmse7_matrx(i,j,1))^2; rmse8_matrx(i,j,3) = (rmse8_matrx(i,j,2) - rmse8_matrx(i,j,1))^2; rmse9_matrx(i,j,3) = (rmse9_matrx(i,j,2) - rmse9_matrx(i,j,1))^2; rmse10_matrx(i,j,3) = (rmse10_matrx(i,j,2) - rmse10_matrx(i,j,1))^2; rmse11_matrx(i,j,3) = (rmse11_matrx(i,j,2) - rmse11_matrx(i,j,1))^2; rmse12_matrx(i,j,3) = (rmse12_matrx(i,j,2) - rmse12_matrx(i,j,1))^2; rmse13_matrx(i,j,3) = (rmse13_matrx(i,j,2) - rmse13_matrx(i,j,1))^2; rmse14_matrx(i,j,3) = (rmse14_matrx(i,j,2) - rmse14_matrx(i,j,1))^2; rmse15_matrx(i,j,3) = (rmse15_matrx(i,j,2) - rmse15_matrx(i,j,1))^2; rmse16_matrx(i,j,3) = (rmse16_matrx(i,j,2) - rmse16_matrx(i,j,1))^2; %rmse17_matrx(i,j,3) = (rmse17_matrx(i,j,2) - rmse17_matrx(i,j,1))^2; %rmse18_matrx(i,j,3) = (rmse18_matrx(i,j,2) - rmse18_matrx(i,j,1))^2; %rmse19_matrx(i,j,3) = (rmse19_matrx(i,j,2) - rmse19_matrx(i,j,1))^2; %rmse20_matrx(i,j,3) = (rmse20_matrx(i,j,2) - rmse20_matrx(i,j,1))^2; end; end; sumse1 = sum(sum(rmse1_matrx(:,:,3))); sumse2 = sum(sum(rmse2_matrx(:,:,3))); sumse3 = sum(sum(rmse3_matrx(:,:,3))); sumse4 = sum(sum(rmse4_matrx(:,:,3))); sumse5 = sum(sum(rmse5_matrx(:,:,3))); sumse6 = sum(sum(rmse6_matrx(:,:,3))); sumse7 = sum(sum(rmse7_matrx(:,:,3))); sumse8 = sum(sum(rmse8_matrx(:,:,3))); sumse9 = sum(sum(rmse9_matrx(:,:,3))); sumse10 = sum(sum(rmse10_matrx(:,:,3))); sumse11 = sum(sum(rmse11_matrx(:,:,3))); sumse12 = sum(sum(rmse12_matrx(:,:,3))); sumse13 = sum(sum(rmse13_matrx(:,:,3))); sumse14 = sum(sum(rmse14_matrx(:,:,3))); sumse15 = sum(sum(rmse15_matrx(:,:,3))); sumse16 = sum(sum(rmse16_matrx(:,:,3))); %sumse17 = sum(sum(rmse17_matrx(:,:,3)));

58

%sumse18 = sum(sum(rmse18_matrx(:,:,3))); %sumse19 = sum(sum(rmse19_matrx(:,:,3))); %sumse20 = sum(sum(rmse20_matrx(:,:,3))); totsumse = sumse1+sumse2+sumse3+sumse4... +sumse5+sumse6+sumse7+sumse8 ... +sumse9+sumse10+sumse11+sumse12+sumse13+sumse14+sumse15+sumse16; % ... %+sumse17+sumse18+sumse19+sumse20; totsumse %tmse = totsumse/(256*256*16); %trmse = sqrt(tmse); %prmse = (100/255)*trmse % Calculation of PC compressed file sizes %stot = 114688; %CR = 16; %oh = 4620; %cvar1 = 1039.48; %cvar2 = 162.40; %cvar3 = 34.64; %cvar4 = 4.46; %cvar5 = 2.61; %cvar6 = 1.73; %cvar7 = 1.01; %x1 = (stot-

oh)/(1+sqrt(cvar2/cvar1)+sqrt(cvar3/cvar1)+sqrt(cvar4/cvar1)+sqrt(cvar5/cvar1)+sqrt(cvar6/cvar1)+sqrt(cvar7/cvar1))

%x2 = (stot-oh)/(1+sqrt(cvar1/cvar2)+sqrt(cvar3/cvar2)+sqrt(cvar4/cvar2)+sqrt(cvar5/cvar2)+sqrt(cvar6/cvar2)+sqrt(cvar7/cvar2))





%x7 =

59

%(stot-oh)/(1+sqrt(cvar1/cvar7)+sqrt(cvar2/cvar7)+sqrt(cvar3/cvar7)+sqrt(cvar4/cvar7)+sqrt(cvar5/cvar7)+sqrt(cvar6/cvar1))

function x = ReadRawImage(filename,width,height) % Read a RAW DATA format image written by IPLAB. % The raw image data is written as a series of unsigned % eight-bit integers in row first order. % Open file fid = fopen(filename); % Read in the data. x = fread(fid, [width,height], 'uint8'); % Close file fclose(fid); function x = ReadRawImage2(filename, width, height) % Read a RAW DATA format image written by IPLAB. % The raw image data is written as a series of unsigned % eight-bit integers in row first order. % Open file id = fopen(filename,'r'); % Read in the data. t = fread(id, [width,height],'uint8'); %x=uint8(zeros(height,width)); x = double(t); %for q=1:height % for w=1:width % x(q,w)=double(t(q,w)); % end %end % Close file fclose(id);

60

% This function takes in spectral images and classifies pixels into one of 6 % classes via an unsupervised minimum Euclidean distance and % General Hill Climb approach. The result is a thematic map. function themmap = isodata_mindist2(files,rws,clmns,M) M(:,:,files+1)=zeros(rws,clmns); cntr1=[]; cntr2=[]; cntr3=[]; cntr4=[]; cntr5=[]; cntr6=[]; for f=1:files cntr1=cat(2,cntr1,M(195,45,f)); cntr2=cat(2,cntr2,M(185,167,f)); cntr3=cat(2,cntr3,M(2,95,f)); cntr4=cat(2,cntr4,M(205,20,f)); cntr5=cat(2,cntr5,M(60,175,f)); cntr6=cat(2,cntr6,M(180,25,f)); end x=1; %count = 262144; %while count>=5243 | x==10 limit = 180000000; % ~((1.8*(255/100))^2)*(256^2)*144; %limit=100000000; sse=limit+1; ssev=[]; while ((sse >= limit) && (x <= 20)) %for x=1:2 count = 0; new_cntr1 = zeros(1,files); new_cntr2 = zeros(1,files); new_cntr3 = zeros(1,files); new_cntr4 = zeros(1,files); new_cntr5 = zeros(1,files); new_cntr6 = zeros(1,files); new_cntr1_count=0; new_cntr2_count=0; new_cntr3_count=0; new_cntr4_count=0; new_cntr5_count=0;

61

new_cntr6_count=0; pixelvlue_to_cntr1=0; pixelvlue_to_cntr2=0; pixelvlue_to_cntr3=0; pixelvlue_to_cntr4=0; pixelvlue_to_cntr5=0; pixelvlue_to_cntr6=0; sse = 0; for i=1:rws for j=1:clmns temp=M(i,j,files+1); for fcnt=1:files pixelvlue_to_cntr1 = pixelvlue_to_cntr1+(M(i,j,fcnt) - cntr1(fcnt))^2; pixelvlue_to_cntr2 = pixelvlue_to_cntr2+(M(i,j,fcnt) - cntr2(fcnt))^2; pixelvlue_to_cntr3 = pixelvlue_to_cntr3+(M(i,j,fcnt) - cntr3(fcnt))^2; pixelvlue_to_cntr4 = pixelvlue_to_cntr4+(M(i,j,fcnt) - cntr4(fcnt))^2; pixelvlue_to_cntr5 = pixelvlue_to_cntr5+(M(i,j,fcnt) - cntr5(fcnt))^2; pixelvlue_to_cntr6 = pixelvlue_to_cntr6+(M(i,j,fcnt) - cntr6(fcnt))^2; end pixelvlue_to_cntr1 = sqrt(pixelvlue_to_cntr1); pixelvlue_to_cntr2 = sqrt(pixelvlue_to_cntr2); pixelvlue_to_cntr3 = sqrt(pixelvlue_to_cntr3); pixelvlue_to_cntr4 = sqrt(pixelvlue_to_cntr4); pixelvlue_to_cntr5 = sqrt(pixelvlue_to_cntr5); pixelvlue_to_cntr6 = sqrt(pixelvlue_to_cntr6); pixelvlue_distcs = [pixelvlue_to_cntr1,pixelvlue_to_cntr2,pixelvlue_to_cntr3,pixelvlue_to_cntr4,pixelvlue_to_cntr5,pixelvlue_to_cntr6]; [r,c]= min(pixelvlue_distcs); M(i,j,files+1) = c; % test=M(i,j,4); if temp ~= M(i,j,files+1) count = count + 1; end switch c case 1 for fcnt1=1:files new_cntr1(fcnt1)=new_cntr1(fcnt1)+M(i,j,fcnt1); end

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new_cntr1_count=new_cntr1_count+1; sse = sse + pixelvlue_to_cntr1^2; case 2 for fcnt2=1:files new_cntr2(fcnt2)=new_cntr2(fcnt2)+M(i,j,fcnt2); end new_cntr2_count=new_cntr2_count+1; sse = sse + pixelvlue_to_cntr2^2; case 3 for fcnt3=1:files new_cntr3(fcnt3)=new_cntr3(fcnt3)+M(i,j,fcnt3); end new_cntr3_count=new_cntr3_count+1; sse = sse + pixelvlue_to_cntr3^2; case 4 for fcnt4=1:files new_cntr4(fcnt4)=new_cntr4(fcnt4)+M(i,j,fcnt4); end new_cntr4_count=new_cntr4_count+1; sse = sse + pixelvlue_to_cntr4^2; case 5 for fcnt5=1:files new_cntr5(fcnt5)=new_cntr5(fcnt5)+M(i,j,fcnt5); end new_cntr5_count=new_cntr5_count+1; sse = sse + pixelvlue_to_cntr5^2; case 6 for fcnt6=1:files new_cntr6(fcnt6)=new_cntr6(fcnt6)+M(i,j,fcnt6); end new_cntr6_count=new_cntr6_count+1; sse = sse + pixelvlue_to_cntr6^2; end end end ssev=cat(2,ssev,sse); if sse <= limit; break;

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end if new_cntr1_count ~= 0 for flc1=1:files cntr1(flc1)=new_cntr1(flc1)/new_cntr1_count; end %clear cenroids to calculate new centroids end if new_cntr2_count ~= 0 for flc2=1:files cntr2(flc2)=new_cntr2(flc2)/new_cntr2_count; end end if new_cntr3_count ~= 0 for flc3=1:files cntr3(flc3)=new_cntr3(flc3)/new_cntr3_count; end end if new_cntr4_count ~= 0 for flc4=1:files cntr4(flc4)=new_cntr4(flc4)/new_cntr4_count; end end if new_cntr5_count ~= 0 for flc5=1:files cntr5(flc5)=new_cntr5(flc5)/new_cntr5_count; end end if new_cntr6_count ~= 0 for flc6=1:files cntr6(flc6)=new_cntr6(flc6)/new_cntr6_count; end end x=x+1; end themmap=M(:,:,files+1);

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% This code reads in the pixel values of two thematic maps and produces % a confusion matrix files=144; rws=256; clmns=256; M=zeros(rws,clmns); %Mb=zeros(rws,clmns); flindx=[1:files]; for f1=1:files if flindx(f1) < 10 file=cat(2,'C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M',char(48+flindx(f1))); file=cat(2,file,'.raw'); M(:,:,f1)=ReadRawImage2(file,clmns,rws); elseif flindx(f1) >= 10 && flindx(f1) < 100 % if (f1 <= filemax) file=cat(2,'C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M',char(48+floor(flindx(f1)/10))); file=cat(2,file,char(48+mod(flindx(f1),10))); file=cat(2,file,'.raw'); M(:,:,f1)=ReadRawImage2(file,clmns,rws); % else % file=cat(2,'C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M',char(48+floor(f1*2/10))); % file=cat(2,file,char(48+mod((f1*2),10))); % file=cat(2,file,'.raw'); % Mb(:,:,((f1*2)-filemax))=ReadRawImage2(file,clmns,rws); % end else % if (f1 <= filemax) file=cat(2,'C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M',char(48+floor(flindx(f1)/100))); file=cat(2,file,char(48+floor((flindx(f1)-((floor(flindx(f1)/100))*100))/10))); file=cat(2,file,char(48+mod(flindx(f1),10))); file=cat(2,file,'.raw'); M(:,:,f1)=ReadRawImage2(file,clmns,rws); % else % file=cat(2,'C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M',char(48+floor(f1*2/100))); % file=cat(2,file,floor((f1*2-floor(f1*2/100)*100)/10)); % file=cat(2,file,char(48+mod((f1*2),10))); % file=cat(2,file,'.raw'); % Mb(:,:,((f1*2)-filemax))=ReadRawImage2(file,clmns,rws);

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% end end end o_thmp=isodata_mindist2(files,rws,clmns,M); %% %files=144; %rws=256; %clmns=256; MR=zeros(rws,clmns); %Mb=zeros(rws,clmns); %flsusd=files; flindx=[1:files]; for f1=1:files if flindx(f1) < 10 file=cat(2,'C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M',char(48+flindx(f1))); file=cat(2,file,'R.raw'); MR(:,:,f1)=ReadRawImage2(file,clmns,rws); elseif flindx(f1) >= 10 && flindx(f1) < 100 % if (f1 <= filemax) file=cat(2,'C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M',char(48+floor(flindx(f1)/10))); file=cat(2,file,char(48+mod(flindx(f1),10))); file=cat(2,file,'R.raw'); MR(:,:,f1)=ReadRawImage2(file,clmns,rws); % else % file=cat(2,'C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M',char(48+floor(f1*2/10))); % file=cat(2,file,char(48+mod((f1*2),10))); % file=cat(2,file,'.raw'); % Mb(:,:,((f1*2)-filemax))=ReadRawImage2(file,clmns,rws); % end else % if (f1 <= filemax) file=cat(2,'C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M',char(48+floor(flindx(f1)/100))); file=cat(2,file,char(48+floor((flindx(f1)-((floor(flindx(f1)/100))*100))/10))); file=cat(2,file,char(48+mod(flindx(f1),10))); file=cat(2,file,'R.raw'); MR(:,:,f1)=ReadRawImage2(file,clmns,rws); % else

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% file=cat(2,'C:\Users\Seton\Desktop\MS_Thesis\PCAcode\M',char(48+floor(f1*2/100))); % file=cat(2,file,floor((f1*2-floor(f1*2/100)*100)/10)); % file=cat(2,file,char(48+mod((f1*2),10))); % file=cat(2,file,'.raw'); % Mb(:,:,((f1*2)-filemax))=ReadRawImage2(file,clmns,rws); % end end end r_thmp=isodata_mindist2(files,rws,clmns,MR); %% classes=6; conf_matrix = zeros(classes,classes); colum_sum = zeros(1,classes); row_sum = zeros(1,classes); overall_accuracy=0; err_of_inclusion=0; err_of_exclusion=0; diagonal_sum=0; total=0; temp=0; erexclud=zeros(1,classes); erinclud=zeros(1,classes); overall_acc=0; for i=1:rws for j=1:clmns conf_matrix(r_thmp(i,j),o_thmp(i,j))=conf_matrix(r_thmp(i,j),o_thmp(i,j))+1; end end for i=1:classes colum_sum(i) = sum(conf_matrix(:,i)); row_sum(i) = sum(conf_matrix(i,:)); diagonal_sum=diagonal_sum+conf_matrix(i,i); erexclud(i)=conf_matrix(i,i)/colum_sum(i); erinclud(i)=conf_matrix(i,i)/row_sum(i); end; total=sum(colum_sum) erexclud erinclud

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overall_acc=diagonal_sum/total conf_matrix figure(1); mesh(o_thmp); figure(2); mesh(r_thmp);

hyperspectral image compression using a two-stage klt

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